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1. The remainder when 10¹⁰ + 10¹⁰⁰ + 10¹⁰⁰⁰ + . . . . . . + 10^1000000000 is divided by 7 is

A. 0

B. 1

C. 2

D. 5

Solution:

Number of terms in the series = 10.
(We can get it easily by pointing the number of zeros in power of terms.
In 1^stterm number of zero is 1, 2^nd term 2, and 3^rd term 3 and so on.)

\frac{10^{10}}{7},

Written as,

\frac{{(7 + 3)}^{(4 \times 2 + 2)}}{7}

The remainder will depend on

\frac{3^{2}}{7}

So, remainder will be 2

\begin{aligned} \frac{10^{1000}}{7}, remainder = 2 \\ \frac{10^{10000}}{7}, remainder = 1 \end{aligned}

So, we get alternate 2 and 1 as remainder, five times each.
So, required remainder is given by

\begin{aligned} \frac{(2 + 1 + 2 + 1 + 2 + 1 + 2 + 1 + 2 + 1)}{7} \\ = \frac{15}{7} \end{aligned}

Remainder when 15 is divided by 7 = 1

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